Chapter 8.3, Problem 17P

In this problem, we establish that the local truncation error for the improved Euler formula is proportional to $h^3$. If we assume that the solution $\varphi$ of the initial value problem $\begin{array}{cc}y\text{'}=f(t,y),& y(t_0)=y_0\end{array}$ has derivatives that are continuous through the third order ( $f$ has continuous second partial derivatives), then it follows that

$\varphi (t_n+h)=\varphi (t_n)+\varphi \text{'}(t_n)h+\frac{\varphi \text{'}\text{'}(t_n)}{2!}{h}^2+\frac{\varphi \text{'}\text{'}\text{'}({\bar{t}}_n)}{3!}{h}^3,$

Where $t_n<\overline{t_n}<t_n+h$. Assume that $y_n=\varphi (t_n)$.

a) Show that, for $y_{n+1}$ as given by Eq.(5) ,

$e_{n+1}=\varphi (t_{n+1})-y_{n+1}$

$\begin{array}{cc}& =\frac{\varphi \text{'}\text{'}(t_n)h-\{f[t_n+h,y_n+hf(t_n,y_n)]-f(t_n,y_n)\}}{2!}+\frac{\varphi \text{'}\text{'}\text{'}(t_n)h^3}{3!}\end{array}$ (i)

b) Making use of the facts that $\varphi \text{'}\text{'}(t)=f_t[t,\varphi (t)]+f_y[t,\varphi (t)]\varphi \text{'}(t)$ and that the Taylor approximation with a remainder for a function $F(t,y)$ of two variable is

$F(a+h,b+k)=f(a,b)+F_t(a,b)h+F_y(a,b)h+\frac{1}{2!}(h^2{F}_{tt}+2hk{F}_{ty}+k^2{F}_{yy}){|}_{x=\xi ,y=\eta }^{}$

Where $\xi$ lies between $a$ and $a+h$ and $\eta$ lies between $b$ and $b+k$, show that the first term on the right side of Eq. (i) is proportional to $h^3$ plus higher order terms. This is the desired result.

c) Show that if $f(t,y)$ is linear in $t$ and $y$, then $e_{n+1}=\varphi \text{'}\text{'}\text{'}(t_n)h^3/6,$ where $t_n<\overline{t_n}<t_n+h$.

Hint: What are $f_{tt},f_{ty},$ and $f_{yy}$?